Answer is: option1

\( \text{(A) } -10 \)

Solution:

We are given the function:

\[ f(x) = (x^3 - 2x + 5)(x^{-2} + x^{-1}) \] Using the product rule: \[ f'(x) = (3x^2 - 2)(x^{-2} + x^{-1}) + (x^3 - 2x + 5)(-2x^{-3} - x^{-2}) \] First, compute individual terms:

• \( g(1) = 1^3 - 2(1) + 5 = 4 \)
• \( g'(1) = 3(1)^2 - 2 = 1 \)
• \( h(1) = 1^{-2} + 1^{-1} = 1 + 1 = 2 \)
• \( h'(1) = -2(1)^{-3} - (1)^{-2} = -2 - 1 = -3 \)

Substituting into the equation: \[ f'(1) = (1)(2) + (4)(-3) \] \[ = 2 - 12 = -10 \] The correct answer is: \[ \boxed{-10} \quad \ \]